Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

Let $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \Omega\subset\R^d$, $d\geqslant 2$ be a bounded domain with smooth boundary $\partial\Omega$ and outer normal vector $\nu$. $L_+^\infty(\Omega)$ denotes the subspace of $L^\infty(\Omega)$-functions with positive essential infima. $\newcommand{\Hd}{H_\diamond} \Hd^1(\Omega)$ and $\newcommand{\Ld}{L_\diamond} \Ld^2(\partial \Omega)$ denote the spaces of $H^1$- and $L^2$-functions with vanishing integral mean on $\partial \Omega$.

For $\sigma\in L_+^\infty(\Omega)$, and a relatively open boundary part $\Sigma\subseteq \partial \Omega$, the local NtD operator $\Lambda(\sigma)$ is defined by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Lambda(\sigma):\ L^2_\diamond(\Sigma)\to L^2_\diamond(\Sigma), \quad g\mapsto u^g_\sigma|_{\Sigma}, \nonumber \end{align*}$$

where $u^g_\sigma\in H^1_\diamond(\Omega)$ is the unique solution of

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{math_model} \nabla\cdot\left(\sigma\nabla u^g_\sigma\right) = 0 {{\rm ~in~}}\Omega,\quad \sigma\partial_\nu u_\sigma^g\vert_{\partial\Omega} = \left\{\begin{array}{@{}ll@{}} g & {{\rm ~on~}}\Sigma, \nonumber \\ 0 & {{\rm ~else.}}\end{array}\right. \nonumber \end{align} \tag{ 2 }$$

This is equivalent to the variational formulation that $u^g_\sigma\in H^1_\diamond(\Omega)$ solves

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Hd}{H_\diamond} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \label{EIT_CM_variational} \int_\Omega \sigma \nabla u^g_\sigma \cdot \nabla w \dx = \int_{\Sigma} g w|_{\Sigma}\dx[s] \quad {{\rm ~for~all~}} w\in \Hd^1(\Omega). \nonumber \end{align} \tag{ 3 }$$

It is well known and easily shown that $\Lambda(\sigma)$ is compact and self-adjoint.

We will consider conductivities that are a priori known to belong to a finite dimensional set of piecewise-analytic functions and that are bounded from above and below by a priori known constants. To that end, we first define piecewise-analyticity as in [46, definition 2.1]:

Definition 2.1. 

• (a)  
  A subset $\Gamma\subseteq \partial O$ of the boundary of an open set $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} O\subseteq \R^n$ is called a smooth boundary piece if it is a $C^\infty$-surface and $O$ lies on one side of it, i.e. if for each $z\in \Gamma$ there exists a ball $\newcommand{\e}{{\rm e}} B_\epsilon(z)$ and a function $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \gamma\in C^\infty(\R^{n-1},\R)$ such that upon relabeling and reorienting

  $$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\Gamma=\partial O\cap B_\epsilon(z)=\{x\in B_\epsilon(z) \; | \; x_n=\gamma(x_1,\ldots,x_{n-1}) \},\nonumber \\ &O\cap B_\epsilon(z)=\{x\in B_\epsilon(z) \; | \; x_n>\gamma(x_1,\ldots,x_{n-1}) \}. \nonumber \end{align*}$$
• (b)  
  $O$ is said to have smooth boundary if $\partial O$ is a union of smooth boundary pieces. $O$ is said to have piecewise smooth boundary if $\partial O$ is a countable union of the closures of smooth boundary pieces.
• (c)  
  A function $\kappa \in L^\infty(\Omega)$ is called piecewise analytic if there exist finitely many pairwise disjoint subdomains $O_1,\ldots,O_M\subset \Omega$ with piecewise smooth boundaries, such that $\overline{\Omega}= \overline{O_1\cup \ldots \cup O_M}$, and $\kappa|_{O_m}$ has an extension which is (real-)analytic in a neighborhood of $\overline{O_m}$, $m=1,\ldots,M$.

Note that (to the knowledge of the author), it is not clear whether the sum of two piecewise-analytic functions is always piecewise-analytic, i.e. whether the set of piecewise-analytic functions is a vector space. However, this can be guaranteed with a slightly stronger definition of piecewise analyticity (see [67, lemma 1]). Moreover, finite-dimensional vector spaces of piecewise-analytic functions (or subsets thereof) naturally arise as parameter spaces for the inverse conductivity problem, e.g. when we fix a partition of the imaging domain $\Omega$ into a finite number of subdomains (e.g. triangles, pixels, or voxels) and the conductivity is assumed to be a polynomial of fixed maximal order on each of these subdomains. Therefore, we make the following definition:

Definition 2.2. A set $\newcommand{\FF}{\mathcal{F}} \FF\subseteq L^\infty(\Omega)$ is called a finite-dimensional subset of piecewise-analytic functions if its linear span

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\psan}{\mathrm{span}\,} \newcommand{\re}{{\rm Re}} \newcommand{\FF}{\mathcal{F}} \newcommand{\N}{{\mathbb {N}}} \newcommand{\R}{{\mathbb {R}}} \displaystyle \psan \FF=\left\{\sum_{j=1}^k \lambda_j \, f_j:\ k\in \N,\ \lambda_j\in \R,\ f_j\in \FF\right\}\subseteq L^\infty(\Omega) \nonumber \end{align*}$$

contains only piecewise-analytic functions and $\newcommand{\FF}{\mathcal{F}} \mathrm{dim}({\rm span}~ \FF)<\infty$.

Given a finite-dimensional subset $\newcommand{\FF}{\mathcal{F}} \FF$ of piecewise analytic functions and two numbers $b>a>0$, we denote the set

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\FF}{\mathcal{F}} \displaystyle \FF_{[a,b]}:=\{\sigma\in \FF:\ a\leqslant \sigma(x)\leqslant b{{\rm ~for~all~}} x\in \Omega\}. \nonumber \end{align*}$$

Throughout this paper, the domain $\Omega$, the finite-dimensional subset $\newcommand{\FF}{\mathcal{F}} \FF$ and the bounds $b>a>0$ are fixed, and the constants in the Lipschitz stability results will depend on them.

Our first result shows Lipschitz stability for the inverse conductivity problem in $\newcommand{\FF}{\mathcal{F}} \FF_{[a,b]}$ when the complete infinite-dimensional NtD-operator is measured.

Theorem 2.3. There exists $c>0$ such that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\FF}{\mathcal{F}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{\Lambda(\sigma_1)-\Lambda(\sigma_2)}_{\LL(\Ld^2(\Sigma))}\geqslant c \norm{\sigma_1-\sigma_2}_{L^\infty(\Omega)} \quad {\rm for~all }~ \sigma_1,\sigma_2\in \FF_{[a,b]}. \nonumber \end{align*}$$

Proof. Theorem 2.3 will be proven in section 2.3. □

We then turn to the question whether $\newcommand{\FF}{\mathcal{F}} \sigma\in \FF_{[a,b]}$ is already uniquely determined by finitely many boundary measurements in the continuum model. For a (finite- or infinite-dimensional) subspace $\newcommand{\Ld}{L_\diamond} G\subseteq \Ld^2(\Sigma)$ we denote by

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \displaystyle P_G:\ \Ld^2(\Sigma) \to G, \quad P_G g=\left\{\begin{array}{@{}ll@{}} g & {{\rm ~if~}} g\in G, \nonumber \\ 0 & {{\rm ~if~}} g\in G^\perp, \end{array} \right. \nonumber \end{align*}$$

the orthogonal projection operator on $G$ with respect to the $L^2$-scalar product

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \label{L2_scalar_product} \langle g,h\rangle:=\int_{\Sigma} gh \dx[s] \quad {{\rm ~for~all~}} g,h\in L^2(\Sigma). \nonumber \end{align} \tag{ 4 }$$

Clearly, $P_G=P_G^*$, and if $G$ is finite dimensional with a basis $\newcommand{\re}{{\rm Re}} \newcommand{\psan}{\mathrm{span}\,} G=\psan(g_1,\ldots,g_n)$ then measurements of

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \langle g_j,\Lambda(\sigma)g_k\rangle \quad j,k=1,\ldots,n \nonumber \end{align*}$$

determine the Galerkin projection of the NtD operator $P_{G} \Lambda(\sigma) {P_G}$, so that this can be regarded as a model for finitely many voltage/current measurements in the continuum model.

Our next result shows that this uniquely determines $\newcommand{\FF}{\mathcal{F}} \sigma\in \FF_{[a,b]}$ (with Lipschitz stability) if the space $G$ is large enough.

Theorem 2.4. For each sequence of subspaces

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \newcommand{\N}{{\mathbb {N}}} \displaystyle G_1\subseteq G_2\subseteq\ldots \Ld^2(\Sigma)\quad {\rm with } \quad \overline{\bigcup_{n\in \N} G_n}=\Ld^2(\Sigma) \nonumber \end{align*}$$

there exists $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} N\in \N$, and $c>0$ such that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{P_{G_n} \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right) {P_{G_n}} }_{\LL(\Ld^2(\Sigma))} \geqslant c \norm{\sigma_1-\sigma_2}_{L^\infty(\Omega)} \nonumber \end{align*}$$

for all $\newcommand{\FF}{\mathcal{F}} \sigma_1,\sigma_2\in \FF_{[a,b]}$, and all $n\geqslant N$.

In particular, this implies that for all $\newcommand{\FF}{\mathcal{F}} \sigma_1,\sigma_2\in \FF_{[a,b]}$ and all $n\geqslant N$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle P_{G_n} \Lambda(\sigma_1) {P_{G_n}}=P_{G_n} \Lambda(\sigma_2) {P_{G_n}} \quad {\rm if~and~only~if } \quad \sigma_1=\sigma_2. \nonumber \end{align*}$$

Proof. Theorem 2.4 will be proven in section 2.3. □

In this subsection, we summarize some known results from the literature, that we will use to prove theorem 2.3 and 2.4. As defined in (4), $\langle \cdot,\cdot \rangle$ always denotes the $L^2(\Sigma)$-scalar product, and $u^g_\sigma\in H^1_\diamond(\Omega)$ denotes the solution of (2) with conductivity $\sigma\in L^\infty_+(\Omega)$ and Neumann data $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\Sigma)$ in the following.

Our first tool is that the NtD operator is continuously Fréchet differentiable with respect to the conductivity.

Lemma 2.5. 

• (a)  
  The mapping

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \displaystyle \Lambda:\ L^\infty_+(\Omega)\to \LL(\Ld^2(\Sigma)),\quad \sigma\mapsto \Lambda(\sigma) \nonumber \end{align*}$$

  is Fréchet differentiable. Its derivative is given by

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \displaystyle \label{Diff_CM} \Lambda'(\sigma)\in \LL(L^\infty(\Omega), \LL(\Ld^2(\Sigma))), \quad \left(\Lambda'(\sigma)\kappa\right)g=v|_{\Sigma}, \nonumber \end{align} \tag{ 5 }$$

  where $\newcommand{\Hd}{H_\diamond} v\in \Hd^1(\Omega)$ solves

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Hd}{H_\diamond} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \int_\Omega \sigma \nabla v\cdot \nabla w \dx = - \int_\Omega \kappa \nabla u_\sigma^{g} \cdot \nabla w \dx \quad {\rm for~all }~w\in \Hd^1(\Omega). \nonumber \end{align*}$$
• (b)  
  For all $\sigma\in L^\infty_+(\Omega)$ and $\kappa\in L^\infty(\Omega)$ the operator $\newcommand{\Ld}{L_\diamond} \newcommand{\LL}{\mathcal{L}} \Lambda'(\sigma)\kappa\in \LL(\Ld^2(\Sigma))$ is self-adjoint and compact, and it fulfills

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \langle \left(\Lambda'(\sigma)\kappa\right) g,h\rangle {=\int_\Omega \sigma \nabla u_\sigma^h \cdot \nabla v \dx }= -\int_\Omega \kappa \nabla u_\sigma^g \cdot \nabla u_\sigma^h \dx, \nonumber \end{align*}$$

  for all $\newcommand{\Ld}{L_\diamond} g,h\in \Ld^2(\Sigma)$.
• (c)  
  The mapping

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \displaystyle \Lambda':\ L^\infty_+(\Omega)\to \LL(L^\infty(\Omega), \LL(\Ld^2(\Sigma))),\quad \sigma\mapsto \Lambda'(\sigma) \nonumber \end{align*}$$

  is continuous.

Proof. This follows from the variational formulation of the conductivity equation (3), see e.g. [70, section 2] or [30, appendix B]. □

Our next tool is a monotonicity relation between the NtD-operator and its derivative that goes back to Ikehata, Kang, Seo, and Sheen [53, 59], and has been used in several other works, see the list of works on monotonicity-based methods cited in the introduction.

Lemma 2.6. For all $\sigma_1,\sigma_2\in L^\infty_+(\Omega)$ and $g\in L^2_\diamond(\Sigma)$, it holds that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\R}{{\mathbb {R}}} \displaystyle \langle \left(\Lambda'(\sigma_2)(\sigma_1-\sigma_2)\right) g,g\rangle &=\int_\Omega(\sigma_2-\sigma_1) |\nabla u_{\sigma_2}^g|^2 \dx\nonumber \\ &\leqslant \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle.\label{eq:monotony_inequality} \nonumber \end{align} \tag{ 6 }$$

Proof. See, e.g. [45, lemma 2.1]. □

The energy terms $|\nabla u_\sigma^g|^2$ in the monotonicity estimate can be controlled using the technique of localized potentials [32]. Roughly speaking, the energy $|\nabla u_\sigma^g|^2$ can be made arbitrarily large in a subset $D_1\subseteq \Omega$ without making it large in another subset $D_2\subseteq \Omega$ whenever $D_1$ can be reached from the boundary without passing $D_2$.

To formulate this rigorously, we adopt the notation from [46, definitions 2.2 and 2.3] and denote by $\newcommand{\inn}{\mathrm{int}\,} \inn D$ the topological interior of a subset $D\subseteq \overline\Omega$, and by $\newcommand{\out}{\mathrm{out}_{\Sigma}\,} \out D$ its outer hull, i.e.

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\out}{\mathrm{out}_{\Sigma}\,} \displaystyle \out D:=\overline \Omega \setminus \bigcup \left\{U\subseteq \overline \Omega:\ U~{{\rm rel.~open,}}~U \cap \Omega ~{{\rm connected}}, \ U\cap \Sigma\neq \emptyset \right\}. \nonumber \end{align*}$$

With this notation, we have the following localized potentials result:

Lemma 2.7. Let $\sigma\in L^\infty_+(\Omega)$ be piecewise analytic and let $D_1,D_2\subseteq\overline\Omega$ be two measurable sets with

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\inn}{\mathrm{int}\,} \newcommand{\out}{\mathrm{out}_{\Sigma}\,} \displaystyle \inn D_1 \nsubseteq \out D_2. \nonumber \end{align*}$$

Then there exists a sequence of currents $(g_m)_{m\in\mathbb {N}}\subset L^2_\diamond(\Sigma)$ such that the corresponding solutions $\newcommand{\Hd}{H_\diamond} (u_\sigma^{g_m})_{m\in \mathbb {N}}\subset \Hd^1(\Omega)$ fulfill

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \lim_{m\rightarrow\infty}\int_{D_1}\left|\nabla u_\sigma^{g_m}\right|^2\,\mathrm{d}x=\infty\quad {\rm and}\quad \lim_{m\rightarrow\infty}\int_{D_2}\left|\nabla u_\sigma^{g_m}\right|^2\,\mathrm{d}x=0. \nonumber \end{align*}$$

proof □

We will also need the following definiteness property of piecewise-analytic functions from [46]:

Lemma 2.8. Let $\newcommand{\e}{{\rm e}} 0\not\equiv \kappa\in L^\infty(\Omega)$ be piecewise-analytic. Then there exist two sets

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\inn}{\mathrm{int}\,} \newcommand{\out}{\mathrm{out}_{\Sigma}\,} \displaystyle D_1=\inn D_1\quad {\rm and } \quad D_2=\out D_2 \nonumber \end{align*}$$

with

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\inn}{\mathrm{int}\,} \newcommand{\out}{\mathrm{out}_{\Sigma}\,} \displaystyle D_1=\inn D_1 \nsubseteq \out D_2=D_2 \nonumber \end{align*}$$

and either

• (i)  
  $\kappa|_{\Omega\setminus D_2}\geqslant 0$ and $\kappa|_{D_1}\in L_+^\infty(D_1)$, or
• (ii)  
  $\kappa|_{\Omega\setminus D_2}\leqslant 0$ and $-\kappa|_{D_1}\in L_+^\infty(D_1)$.

proof □

We can now prove theorems 2.3 and 2.4. For the sake of brevity, we write $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \norm{\cdot}$ for $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \newcommand{\Ld}{L_\diamond} \newcommand{\LL}{\mathcal{L}} \norm{\cdot}_{\LL(\Ld^2(\Sigma))}$, $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \norm{\cdot}_{L^\infty(\Omega)}$ and $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \newcommand{\Ld}{L_\diamond} \norm{\cdot}_{\Ld^2(\Sigma)}$ throughout this subsection.

We follow the approach in [41] and first use the monotonicity relation in lemma 2.6 to bound the difference of the non-linear NtD operators by an expression containing their linearized counterparts.

Lemma 2.9. For all $\newcommand{\FF}{\mathcal{F}} \sigma_1,\sigma_2\in \FF_{[a,b]}$ with $\newcommand{\e}{{\rm e}} \sigma_1\not\equiv \sigma_2$,

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \frac{\norm{\Lambda(\sigma_1)-\Lambda(\sigma_2)}} {\norm{\sigma_1-\sigma_2}} \geqslant \inf_{(\tau_1,\tau_2,\kappa)\atop \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK} \ \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\tau_1,\tau_2,\kappa,g), \nonumber \end{align*}$$

where $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\Ld}{L_\diamond} f:\ L^\infty_+(\Omega)\times L^\infty_+(\Omega)\times L^\infty(\Omega)\times \Ld^2(\Sigma)\to \R$ is defined by

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \displaystyle f(\tau_1,\tau_2,\kappa,g):=\max \left\{\Langle \left(\Lambda'(\tau_1) \kappa\right) g,g \Rangle, -\Langle \left(\Lambda'(\tau_2) \kappa\right) g,g \Rangle\right\}, \nonumber \end{align*}$$

and $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} \newcommand{\re}{{\rm Re}} \newcommand{\psan}{\mathrm{span}\,} \KK:=\{\kappa\in \psan\FF:\ \norm{\kappa}=1\}$.

Proof. The NtD-operators are self-adjoint so that for all $\sigma_1,\sigma_2\in L^\infty(\Omega)$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\Ld}{L_\diamond} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{\Lambda(\sigma_1)-\Lambda(\sigma_2)}=\sup_{g\in \Ld^2(\Sigma), \norm{g}=1} \left| \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle \right|. \nonumber \end{align*}$$

Using the monotonicity inequality in lemma 2.6 also with interchanged roles of $\sigma_1$ and $\sigma_2$, we obtain that for all $\sigma_1,\sigma_2\in L^\infty_+(\Omega)$, $\newcommand{\e}{{\rm e}} \sigma_1\not\equiv \sigma_2$, and all $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\partial \Omega)$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle {\left| \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle \right|}\nonumber \\ = \max \{\Langle g, \left(\Lambda(\sigma_2)-\Lambda(\sigma_1)\right)g \Rangle, \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle\}\nonumber \\ \geqslant \max \{\Langle \Lambda'(\sigma_1) (\sigma_2-\sigma_1) g,g \Rangle, \Langle \Lambda'(\sigma_2) (\sigma_1-\sigma_2) g,g \Rangle\}\nonumber \\ = \norm{\sigma_1-\sigma_2} \max \left\{\Langle \Lambda'(\sigma_1) \frac{\sigma_2-\sigma_1}{\norm{\sigma_1-\sigma_2}} g,g \Rangle, \Langle \Lambda'(\sigma_2) \frac{\sigma_1-\sigma_2}{\norm{\sigma_1-\sigma_2}} g,g \Rangle\right\}\nonumber \\ = \norm{\sigma_1-\sigma_2} f(\sigma_1,\sigma_2,\frac{\sigma_2-\sigma_1}{\norm{\sigma_1-\sigma_2}},g). \nonumber \end{align*}$$

Hence,

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Ld}{L_\diamond} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \frac{\norm{\Lambda(\sigma_1)-\Lambda(\sigma_2)}} {\norm{\sigma_1-\sigma_2}} &= \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} \frac{\left| \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle \right|} {\norm{\sigma_1-\sigma_2}}\nonumber \\ &\geqslant \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\sigma_1,\sigma_2,\frac{\sigma_2-\sigma_1}{\norm{\sigma_1-\sigma_2}},g)\nonumber \\ &\geqslant \inf_{(\tau_1,\tau_2,\kappa)\atop \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK}\ \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\tau_1,\tau_2,\kappa,g). \nonumber \end{align*}$$

□

Now we use a compactness argument to show that the expression in the lower bound in lemma 2.9 attains its minimum.

Lemma 2.10. There exists $\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} (\hat \tau_1,\hat \tau_2,\hat \kappa) \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK$ so that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \inf_{(\tau_1,\tau_2,\kappa)\atop \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK} \ \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\tau_1,\tau_2,\kappa,g) = \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\hat \tau_1,\hat \tau_2,\hat \kappa,g). \nonumber \end{align*}$$

Proof. Since $f$ is continuous by lemma 2.5, the function

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle (\tau_1,\tau_2,\kappa)\mapsto \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\tau_1,\tau_2,\kappa,g) \nonumber \end{align*}$$

is lower semicontinuous and thus attains its minimum over the compact set $\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} \FF_{[a,b]}\times$$\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} \FF_{[a,b]}\times \KK$. □

It remains to show that the minimum attained in lemma 2.10 must be positive. To show that we use the localized potentials from lemma 2.7.

Lemma 2.11. Let $\newcommand{\e}{{\rm e}} 0\not\equiv \kappa\in L^\infty(\Omega)$ be piecewise-analytic. Then at least one of the following two properties holds true:

• (i)  
  For all piecewise analytic $\sigma\in L^\infty_+(\Omega)$ there exists $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\partial \Omega)$ with

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle -\langle \left(\Lambda'(\sigma)\kappa \right) g,g\rangle =\int_\Omega \kappa |\nabla u_{\sigma}^g|^2 \dx >0. \nonumber \end{align*}$$
• (ii)  
  For all piecewise analytic $\sigma\in L^\infty_+(\Omega)$ there exists $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\partial \Omega)$ with

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle -\langle \left(\Lambda'(\sigma)\kappa \right) g,g\rangle =\int_\Omega \kappa |\nabla u_{\sigma}^g|^2 \dx <0. \nonumber \end{align*}$$

Hence, a fortiori,

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\tau_1,\tau_2,\kappa,g)>0 \quad {\rm for~all }~(\tau_1,\tau_2,\kappa)\in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK. \nonumber \end{align*}$$

Proof. Using the definiteness property of piecewise analytic functions from lemma 2.8 we obtain two sets $\newcommand{\inn}{\mathrm{int}\,} D_1=\inn D_1$ and $\newcommand{\out}{\mathrm{out}_{\Sigma}\,} D_2=\out D_2$ with

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\inn}{\mathrm{int}\,} \newcommand{\out}{\mathrm{out}_{\Sigma}\,} \displaystyle D_1=\inn D_1 \nsubseteq \out D_2=D_2 \nonumber \end{align*}$$

and either

• (i)  
  $\kappa|_{\Omega\setminus D_2}\geqslant 0$ and $\kappa|_{D_1}\in L_+^\infty(D_1)$, or
• (ii)  
  $\kappa|_{\Omega\setminus D_2}\leqslant 0$ and $-\kappa|_{D_1}\in L_+^\infty(D_1)$.

Let $(g_m)_{m\in\mathbb {N}}\subset L^2_\diamond(\Sigma)$ be the localized potentials sequence from lemma 2.7. Then, in case (i), we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle {-\langle \left(\Lambda'(\sigma)\kappa \right) g_m,g_m\rangle=\int_\Omega \kappa |\nabla u_\sigma^{g_m}|^2 \dx}\nonumber \\ = \int_{D_1} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx + \int_{D_2} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx \int_{\Omega\setminus (D_1\cup D_2)} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx\nonumber \\ \geqslant \mathrm{ess\,inf}\, \kappa|_{D_1} \int_{D_1} |\nabla u_{\sigma}^{g_m}|^2 \dx - \norm{\kappa}_{L^\infty(D_2)} \int_{D_2} |\nabla u_{\sigma}^{g_m}|^2 \dx\to \infty, \nonumber \end{align*}$$

so that $\newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \int_\Omega \kappa |\nabla u_\sigma^{g_m}|^2 \dx>0$ for sufficiently large $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} m\in \N$.

In case (ii) we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle {-\langle \left(\Lambda'(\sigma)\kappa \right) g_m,g_m\rangle=\int_\Omega \kappa |\nabla u_\sigma^{g_m}|^2 \dx}\nonumber \\ = \int_{D_1} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx + \int_{D_2} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx \int_{\Omega\setminus (D_1\cup D_2)} \kappa |\nabla u_{\sigma}^{g_m}|^2 \dx\nonumber \\ \leqslant -\mathrm{ess\,inf}\, (-\kappa)|_{D_1} \int_{D_1} |\nabla u_{\sigma}^{g_m}|^2 \dx + \norm{\kappa}_{L^\infty(D_2)} \int_{D_2} |\nabla u_{\sigma}^{g_m}|^2 \dx\to -\infty, \nonumber \end{align*}$$

so that $\newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \int_\Omega \kappa |\nabla u_\sigma^{g_m}|^2 \dx<0$ for sufficiently large $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} m\in \N$. □

Remark 2.12. It is known (see e.g. [45, corollary 3.5(b)]) that for all piecewise analytic $\sigma$, the Fréchet derivative $\Lambda'(\sigma)$ is injective on the set of piecewise analytic functions, i.e. $\Lambda'(\sigma)\kappa\neq 0$ for all piecewise analytic $\newcommand{\e}{{\rm e}} 0\not\equiv \kappa\in L^\infty(\Omega)$.

Since $\Lambda'(\sigma)\kappa$ is a compact self-adjoint operator, this means that $\Lambda'(\sigma)\kappa$ must possess either a positive or a negative eigenvalue. Lemma 2.11 can be interpreted in the sense, that for each $\newcommand{\e}{{\rm e}} \kappa\not\equiv 0$ this property is sign-uniform in $\sigma$, i.e. for each $\newcommand{\e}{{\rm e}} \kappa\not\equiv 0$, the operator $\Lambda'(\sigma)\kappa$ either possesses a positive eigenvalue for all $\sigma$, or it possesses a negative eigenvalue for all $\sigma$ (or both properties are fulfilled).

With these preparations we can now show the theorems 2.3 and 2.4.

Proof of theorem 2.3. The assertion follows from lemmas 2.9–2.11 with

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle c:=\sup_{g\in \Ld^2(\Sigma), \norm{g}=1} f(\hat \tau_1,\hat \tau_2,\hat \kappa,g)>0. \nonumber \end{align*}$$

□

Proof of theorem 2.4. Using that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{P_{G_n}\left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right) {P_{G_n}}} {=} \sup_{g\in G_n, \norm{g}=1} \left| \Langle g, \left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right)g \Rangle \right|, \nonumber \end{align*}$$

we obtain as in lemmas 2.9 and 2.10 that for all $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} n\in \N$, there exists $\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} (\hat \tau_1^{(n)},\hat \tau_2^{(n)},\hat \kappa^{(n)})$$\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK$ so that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \label{Galerkin_hilf1} \frac{\norm{P_{G_n}\left(\Lambda(\sigma_1)-\Lambda(\sigma_2)\right) {P_{G_n}}}} {\norm{\sigma_1-\sigma_2}} \geqslant \sup_{g\in G_n, \norm{g}=1} f(\hat \tau_1^{(n)},\hat \tau_2^{(n)},\hat \kappa^{(n)},g). \nonumber \end{align} \tag{ 7 }$$

The right hand side of (7) is monotonically increasing in $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} n\in \N$ since the spaces $G_n$ are nested. Hence, the assertion of theorem 2.4 follows, if we can prove that there exists $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} n\in \N$ with

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \label{sup_larger_zero_Galerkin} \sup_{g\in G_n, \norm{g}=1} f( \tau_1, \tau_2, \kappa,g)>0 \quad {{\rm ~for~all~}} (\tau_1,\tau_2,\kappa)\in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK. \nonumber \end{align} \tag{ 8 }$$

We argue by contradiction and assume that this is not the case. Then there exists a sequence $\newcommand{\KK}{\mathcal{K}} (\tau_1^{(n)},\tau_2^{(n)},\kappa^{(n)})\in {\mathcal F}_{[a,b]}\times {\mathcal F}_{[a,b]}\times \KK$ with

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{g\in G_n, \norm{g}=1} f(\tau_1^{(n)},\tau_2^{(n)},\kappa^{(n)},g)\leqslant 0 \quad {{\rm ~for~all~}} n\in \N, \nonumber \end{align*}$$

which also implies

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{g\in G_m, \norm{g}=1} f(\tau_1^{(n)},\tau_2^{(n)},\kappa^{(n)},g)\leqslant 0 \quad {{\rm ~for~all~}} n\in \N,\ n\geqslant m. \nonumber \end{align*}$$

After passing to a subsequence if necessary, we can assume by compactness that the sequence $(\tau_1^{(n)},\tau_2^{(n)},\kappa^{(n)})$ converges to some element

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \displaystyle (\hat\tau_1,\hat\tau_2,\hat\kappa)\in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK. \nonumber \end{align*}$$

Since, for all $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} m\in \N$, the function

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle (\tau_1,\tau_2,\kappa)\mapsto \sup_{g\in G_m, \norm{g}=1} f(\tau_1,\tau_2,\kappa,g) \nonumber \end{align*}$$

is lower semicontinuous, it follows that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{g\in G_m, \norm{g}=1} f(\hat\tau_1,\hat\tau_2,\hat\kappa,g)\leqslant 0 \quad {{\rm ~for~all~}} m\in \N. \nonumber \end{align*}$$

But, by continuity, this would imply

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \newcommand{\N}{{\mathbb {N}}} \displaystyle f(\hat\tau_1,\hat\tau_2,\hat\kappa,g)\leqslant 0 \quad {{\rm ~for~all~}} g\in \overline{\bigcup_{m\in \N} G_m}=\Ld^2(\Sigma), \nonumber \end{align*}$$

which contradicts lemma 2.11. This shows that (8) must be true for sufficiently large $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} n\in \N$ and thus theorem 2.4 is proven. □

Now we consider the CEM. As before let $\sigma\in L^\infty_+(\Omega)$ denote the conductivity distribution in a smoothly bounded domain $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \Omega\subseteq \R^d$, $d\geqslant 2$. We assume that $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} M\in \N$ open, connected, mutually disjoint electrodes $E_m\subseteq \partial \Omega$, $m=1,\ldots,M$, are attached to the boundary of $\Omega$ with all electrodes having the same contact impedance $z>0$. When a current with strength $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} J_m\in \R$ is driven through the $m$-th electrode (with $\sum_{m=1}^M J_m=0$), the resulting electrical potential $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} (u,U)\in H^1(\Omega)\times \R^M$ solves the following equations:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{CEM1} \nabla \cdot \sigma\nabla u=0 \qquad\quad\qquad\quad\,\,\,\, {{\rm in~}} \Omega, \nonumber \end{align} \tag{ 9 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \label{CEM2} \sigma\partial_\nu u=0 \qquad\qquad\qquad\qquad {{\rm on~}} \partial \Omega\setminus \bigcup \nolimits_{m=1}^M E_m, \nonumber \end{align} \tag{ 10 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{CEM3} u+z \sigma \partial_\nu u=\mathrm{const.}=:U_m \quad ~ {{\rm on~}} E_m,~ m=1,\ldots,M, \nonumber \end{align} \tag{ 11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \label{CEM4} \int_{E_m} \sigma \partial_\nu u|_{E_m}\dx[s] =J_m \qquad\qquad {{\rm on~}} E_m, ~m=1,\ldots,M, \nonumber \end{align} \tag{ 12 }$$

where $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} U=(U_1,\ldots,U_M)\in \R^M$ is a vector containing the electric potentials on the electrodes $E_1,\ldots,E_M$.

It can be shown that (9)–(12) possess a solution $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} (u,U)\in H^1(\Omega)\times \R^m$ and that the solution is unique under the additional gauge (or ground level) condition $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} U\in \Rd^M$, where $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \Rd^M$ is the subspace of vectors in $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \R^m$ with zero mean, see e.g. [82]. We can thus define the $M$-electrode current-to-potential operator

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \displaystyle R_M(\sigma):\ \Rd^M\to \Rd^M:\ I=(I_1,\ldots,I_M) \mapsto U=(U_1,\ldots,U_M), \nonumber \end{align*}$$

where $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} (u,U)\in H^1(\Omega)\times \Rd^m$ solves (9)–(12). Note also that (9)–(12) are equivalent to the variational formulation that $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} (u,U)\in H^1(\Omega)\times \Rd^m$ solves

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \label{CEM_Variational} \int_\Omega \sigma \nabla u \cdot \nabla w \dx + \sum_{m=1}^M \int_{E_m} \frac{1}{z} (u-U_m)(w-W_m) \dx[s] = \sum_{m=1}^M J_m W_m \nonumber \end{align} \tag{ 13 }$$

for all $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} (w,W)\in H^1(\Omega)\times \Rd^m$, see again, [82].

As in the previous section, we will consider conductivities that belong to a finite dimensional subset of piecewise-analytic functions. Additionally, in order to use results from [30] on the approximation properties of the CEM, we assume that the background conductivity is an a priori known smooth function in a fixed neighborhood $O$ of the boundary $\partial \Omega$, i.e. we assume that $\newcommand{\FF}{\mathcal{F}} \FF$ is a finite dimensional subset of piecewise-analytic functions, so that there exists $\sigma_0\in C^\infty({\overline O})$ with $\sigma|_{{O}}=\sigma_0|_{{O}}$ for all $\newcommand{\FF}{\mathcal{F}} \sigma\in \FF$. Together with the assumption of a priori known bounds, we assume (for $b>a>0$)

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\FF}{\mathcal{F}} \displaystyle \sigma\in \FF_{[a,b]}:=\{\sigma\in \FF:\ a\leqslant \sigma(x)\leqslant b{{\rm ~for~all~}} x\in \Omega\}. \nonumber \end{align*}$$

We will show that $R_M(\sigma)$ uniquely determines $\newcommand{\FF}{\mathcal{F}} \sigma\in\FF_{[a,b]}$ (with Lipschitz stability) if, roughly speaking, enough electrodes are being used. To make this statement precise, assume that the number of electrodes is increased so that the electrode configurations fulfill the Hyvönen criteria [30, 52]:

• (H1)  
  For all electrode configurations

  $$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle E_1^{{(M)}},\ldots,E_M^{{(M)}}\subseteq \partial \Omega \nonumber \end{align*}$$

  there exist open, connected, and mutually disjoint sets (called virtual extended electrodes) $\tilde E_m^{(M)}$, $m=1,\ldots,M$ with

  $$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle E_m^{(M)}\subseteq \tilde E_m^{(M)}, \quad \bigcup_{m=1}^M \overline{\tilde E_m^{(M)}}=\partial \Omega, \nonumber \end{align*}$$

  so that

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} \displaystyle h_{M}:=\max_{m=1,\ldots,M} \left\{\sup_{x,y\in \tilde E_m^{(M)}} \mathrm{dist}({x,y})\right\} \to 0 \quad & {{\rm ~for~}} M\to \infty,\nonumber \\ \exists c_{\mathrm{E}}>0:\ \min_{m=1,\ldots,M} \frac{|E_m^{(M)}|}{|\tilde E_m^{(M)}|}\geqslant c_{\mathrm{E}} \quad & {{\rm ~for~all~}} M\in \N. \nonumber \end{align*}$$
• (H2)  
  The operators

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\R}{{\mathbb {R}}} \displaystyle Q_{M}:& \ \R^M\to L^2(\partial \Omega),\quad (J_m)_{m=1}^M \mapsto \sum_{m=1}^M J_m \chi_{\tilde E_m^{(M)}}\nonumber \\ P_{M}:& \ L^2(\partial \Omega)\to \R^M,\quad g\mapsto \left(\frac{1}{|E_m^{(M)}|} \int_{E_m^{(M)}} g \dx[s]\right)_{m=1}^M \nonumber \end{align*}$$

  fulfill that

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \exists C_{\mathrm{E}}>0:\ \norm{(I-Q_{M}P_{M})\,f}_{L^2(\partial \Omega)}\leqslant C_{\mathrm{E}} h_M \inf_{c\in \R} \norm{\,f+c}_{H^1(\partial \Omega)} \nonumber \end{align*}$$

  for all $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} M\in \N$ and all $f\in H^1(\partial \Omega)$.

The first criterion implies the natural assumption that the electrode sizes shrink to zero, but always cover a certain fraction of the boundary. The somewhat technical second criterion can be interpreted as a Poincaré-type inequality that is fulfilled for regular enough electrode shapes, see [30, 52, 69]. Together these criteria guarantee that the electrode measurement approximate all possible continuous measurements in a suitable sense.

Now we can state our main result:

Theorem 3.1. There exists $\newcommand{\e}{{\rm e}} \newcommand{\N}{{\mathbb {N}}} N\in \N$ and $c>0$ such that for all $M\geqslant N$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\FF}{\mathcal{F}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{R_M(\sigma_1) - R_M(\sigma_2)}_{\LL(\Rd^M)}\geqslant c \norm{\sigma_1-\sigma_2}_{L^\infty(\Omega)} \quad {\rm for~all }~ \sigma_1,\sigma_2\in \FF_{[a,b]}. \nonumber \end{align*}$$

In particular, this implies that for all $\newcommand{\FF}{\mathcal{F}} \sigma_1,\sigma_2\in \FF_{[a,b]}$ and $M\geqslant N$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle R_M(\sigma_1) = R_M(\sigma_2) \quad {\rm if~and~only~if }~\quad \sigma_1=\sigma_2. \nonumber \end{align*}$$

Theorem 3.1 will be proven in section 3.3.

The electrode measurements $R_M(\sigma)$ fulfill analogue differentiability and monotonicity properties as the NtD-Operators. In the following $\langle \cdot,\cdot \rangle_M$ denotes the Euclidian scalar product in $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \R^M$. For a vector $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} J=(J_1,\ldots,J_M)\in \Rd^M$, we denote by $u^{(J)}_\sigma\in H^1_\diamond(\Omega)$ the solution of the CEM equations (9)–(12) with conductivity $\sigma\in L^\infty_+(\Omega)$ and electrode currents $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} J_1,\ldots,J_m\in \R$.

Lemma 3.2. 

• (a)  
  The mapping

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \displaystyle R_M:\ L^\infty_+(\Omega)\to \LL(\Rd^M),\quad \sigma\mapsto R_M(\sigma) \nonumber \end{align*}$$

  is Fréchet differentiable. Its derivative is given by

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \displaystyle R_M'(\sigma)\in \LL(L^\infty(\Omega), \LL(\Rd^M)), \quad \left(R_M'(\sigma)\kappa\right) J= V, \nonumber \end{align*}$$

  where $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} (v,V)\in H^1(\Omega)\times \Rd^m$ solves

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle &\int_\Omega \sigma \nabla v \cdot \nabla w \dx + \sum_{m=1}^M \int_{E_m} \frac{1}{z} (v-V_m)(w-W_m) \dx[s]\nonumber \\ &= - \int_\Omega \kappa \nabla u_\sigma^{(J)}\cdot \nabla w \dx \label{Diff_CEM} \nonumber \end{align} \tag{ 14 }$$

  for all $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} (w,W)\in H^1(\Omega)\times \Rd^m$.
• (b)  
  For all $\sigma\in L^\infty_+(\Omega)$ and $\kappa\in L^\infty(\Omega)$ the operator $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\LL}{\mathcal{L}} R_M'(\sigma)\kappa\in \LL(\Rd^M)$ is self-adjoint, and it fulfills

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle \langle \left(R_M'(\sigma)\kappa\right) I,J\rangle_M=-\int_\Omega \kappa \nabla u_\sigma^{(I)} \cdot \nabla u_\sigma^{(J)} \dx, \nonumber \end{align*}$$

  for all $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} I,J\in \Rd^M$.
• (c)  
  The mapping

  $$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \displaystyle R_M':\ L^\infty_+(\Omega)\to \LL(L^\infty(\Omega), \LL(\Rd^M)),\quad \sigma\mapsto R_M'(\sigma) \nonumber \end{align*}$$

  is continuous.

Proof. This follows from the variational formulation of the CEM (13), see e.g. [70, section 2] or [30, appendix B]. □

Lemma 3.3. For all $\sigma_1,\sigma_2\in L^\infty_+(\Omega)$ and $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} J\in \Rd^M$, it holds that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\R}{{\mathbb {R}}} \displaystyle \langle \left(R'(\sigma_2)(\sigma_1-\sigma_2)\right) J,J\rangle_M&= \int_\Omega(\sigma_2-\sigma_1) |\nabla u_{\sigma_2}^{(J)}|^2 \dx\nonumber \\ &\leqslant \Langle \left(R_M(\sigma_1)-R_M(\sigma_2)\right)J,J \Rangle_M. \nonumber \end{align*}$$

proof □

We will also require the following result from Garde and Staboulis [30] that the linearized CEM measurements approximate the linearized NtD operator.

Lemma 3.4. Under the Hyvönen assumptions (H1) and (H2), there exists $C>0$ such that for all $\newcommand{\FF}{\mathcal{F}} \sigma\in \FF_{[a,b]}$ and $\newcommand{\FF}{\mathcal{F}} \newcommand{\re}{{\rm Re}} \newcommand{\psan}{\mathrm{span}\,} \kappa\in \psan \FF$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Ld}{L_\diamond} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{\Lambda'(\sigma)\kappa- L Q_{{M}} \left( R'(\sigma)\kappa\right) Q^*_{{M}} }_{\LL(\Ld^2(\partial \Omega))} \leqslant C h_M \norm{\sigma}_{L^\infty(\Omega)} \norm{\kappa}_{L^\infty(\Omega)}, \nonumber \end{align*}$$

where

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \displaystyle L:\ {L^2(\partial \Omega)\to L^2_\diamond(\partial \Omega)},\quad g \mapsto g-\frac{1}{|\partial \Omega|}\int_{\partial \Omega} g \dx[s], \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\R}{{\mathbb {R}}} \displaystyle Q_M^*:\ L^2(\partial \Omega)\to \R^M,\quad g \mapsto \left(\int_{\tilde E_1^{(M)}} g\dx[s],\ldots,\int_{\tilde E_M^{(M)}} g\dx[s]\right). \nonumber \end{align*}$$

Moreover, for all $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\partial \Omega)$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \displaystyle \Langle L Q_{{M}} \left( R'(\sigma)\kappa\right) Q^*_{{M}} g, g\Rangle = \Langle \left( R'(\sigma)\kappa\right) Q^*_{{M}} g, Q^*_{{M}} g\Rangle_M. \nonumber \end{align*}$$

proof □

Again, for the sake of brevity, we omit norm subscripts when the choice of the norm is clear from the context. As in section 2.3 we obtain from the monotonicity result lemma 3.3 that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{R_M(\sigma_1)-R_M(\sigma_2)} \geqslant \norm{\sigma_1-\sigma_2} \inf_{(\tau_1,\tau_2,\kappa) \atop \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK} \ \sup_{J\in \Rd^M\atop \norm{J}=1} f_M(\tau_1,\tau_2,\kappa,J), \nonumber \end{align*}$$

where $\newcommand{\R}{{\mathbb {R}}} \newcommand{\e}{{\rm e}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} f_M:\ L^\infty_+(\Omega)\times L^\infty_+(\Omega)\times L^\infty(\Omega)\times \Rd^M\to \R$ is defined by

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \displaystyle f_M(\tau_1,\tau_2,\kappa,J):=\max \left\{\Langle \left(R_M'(\tau_1) \kappa\right) J,J \Rangle, -\Langle \left(R_M'(\tau_2) \kappa\right) J,J \Rangle\right\}, \nonumber \end{align*}$$

and $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} \newcommand{\re}{{\rm Re}} \newcommand{\psan}{\mathrm{span}\,} \KK:=\{\kappa\in \psan \FF:\ \norm{\kappa}=1\}$.

We compare this with $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\Ld}{L_\diamond} f:\ L^\infty_+(\Omega)\times L^\infty_+(\Omega)\times L^\infty(\Omega)\times \Ld^2(\partial \Omega)\to \R$,

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \displaystyle f(\tau_1,\tau_2,\kappa,g):=\max \left\{\Langle \left(\Lambda'(\tau_1) \kappa\right) g,g \Rangle, -\Langle \left(\Lambda'(\tau_2) \kappa\right) g,g \Rangle\right\}, \nonumber \end{align*}$$

from the continuum model (see lemma 2.9) with $\Sigma=\partial \Omega$.

For all real numbers $\newcommand{\e}{{\rm e}} \newcommand{\R}{{\mathbb {R}}} r_1,r_2,s_1,s_2\in \R$ one has that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \max\{r_1,s_1\}-\max\{r_2,s_2\} &= \frac{1}{2}\left( r_1+s_1 + |r_1-s_1| - r_2 -s_2 - |r_2-s_2|\right)\nonumber \\ &\leqslant \frac{1}{2}\left( r_1+s_1 + |r_1-s_1 - (r_2 - s_2)| - r_2 -s_2 \right)\nonumber \\ &=\max\{(r_1-r_2),(s_1-s_2) \}. \nonumber \end{align*}$$

Hence, we obtain with lemma 3.4 that for all $\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} (\tau_1,\tau_2,\kappa) \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK$ and $\newcommand{\Ld}{L_\diamond} g\in \Ld^2(\partial \Omega)$ with $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \norm{g}=1$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Rangle}{\right\rangle} \newcommand{\Langle}{\left\langle} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle &{\left| \, f(\tau_1,\tau_2,\kappa,g)-f_M(\tau_1,\tau_2,\kappa,Q_M^* g)\right|}\nonumber \\ &{\leqslant} {\max \left\{\left| \Langle \left( \Lambda'(\tau_1) \kappa - Q_M \left(R_M'(\tau_1) \kappa\right) Q_M^* \right) g,g \Rangle\right|, \right.}\nonumber \\ & {\qquad \qquad \qquad \left. \left| \Langle \left( \Lambda'(\tau_2) \kappa - Q_M \left(R_M'(\tau_2) \kappa\right) Q_M^* \right) g,g \Rangle\right|\right\}}\nonumber \\ & \leqslant Ch_M \norm{\kappa} \norm{g} \max\{\norm{\tau_1},\norm{\tau_2}\}\leqslant C h_M b {,} \nonumber \end{align*}$$

where $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \norm{\kappa}=1$ and $\newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \max\{\norm{\tau_1},\norm{\tau_2}\}\leqslant b$ follows from the definition of $\newcommand{\KK}{\mathcal{K}} \KK$ and $\newcommand{\FF}{\mathcal{F}} \FF_{[a,b]}$.

For all $g\in {L^2(\partial \Omega)}$ we have that

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dx}[1][x]{{\,{{\rm d}} #1}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \norm{Q_M ^* g}^2= \sum_{m=1}^M \left(\int_{\tilde E_m^{(M)}} g \dx[s]\right)^2 \leqslant \sum_{m=1}^M |\tilde E_m^{(M)}| \int_{\tilde E_m^{(M)}} g^2 \dx[s] \leqslant \left| \partial \Omega \right| \norm{g}^2, \nonumber \end{align*}$$

so that we obtain for all $\newcommand{\FF}{\mathcal{F}} \newcommand{\KK}{\mathcal{K}} (\tau_1,\tau_2,\kappa) \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\Ld}{L_\diamond} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{J\in \Rd^M,\atop \norm{J}=1} f_M(\tau_1,\tau_2,\kappa,J) & \geqslant \left| \partial \Omega \right|^{-1} \sup_{g\in \Ld^2(\partial \Omega),\atop \norm{g}=1} f_M(\tau_1,\tau_2,\kappa,Q_M^*g)\nonumber \\ & \geqslant \left| \partial \Omega \right|^{-1} \sup_{g\in \Ld^2(\partial \Omega),\atop \norm{g}=1} f(\tau_1,\tau_2,\kappa,g) - \left| \partial \Omega \right|^{-1} Ch_M b. \nonumber \end{align*}$$

Since the first summand is positive by lemma 2.11, it follows that for sufficiently large numbers of electrodes $M$

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \sup_{J\in \Rd^M,\atop \norm{J}=1} f_M(\tau_1,\tau_2,\kappa,J)>0 \quad {{\rm ~for~all~}} (\tau_1,\tau_2,\kappa) \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK. \nonumber \end{align*}$$

With the same lower semicontinuity and compactness argument as in the continuum model, this yields

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\KK}{\mathcal{K}} \newcommand{\FF}{\mathcal{F}} \newcommand{\Rd}{{\mathbb {R}_\diamond}} \newcommand{\R}{{\mathbb {R}}} \newcommand{\norm}[1]{\hspace{0.2ex} \|#1\| \hspace{0.2ex}} \displaystyle \inf_{(\tau_1,\tau_2,\kappa) \atop \in \FF_{[a,b]}\times \FF_{[a,b]}\times \KK} \ \sup_{J\in \Rd^M,\atop \norm{J}=1} f_M(\tau_1,\tau_2,\kappa,J)>0, \nonumber \end{align*}$$

so that the assertion is proven. □